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theory HoTT_Theorems
imports HoTT
begin
text "A bunch of theorems and other statements for sanity-checking, as well as things that should be automatically simplified.
Things that *should* be automated:
\<bullet> Checking that \<open>A\<close> is a well-formed type, when writing things like \<open>x : A\<close> and \<open>A : U\<close>.
\<bullet> Checking that the argument to a (dependent/non-dependent) function matches the type? Also the arguments to a pair?
"
\<comment> \<open>Turn on trace for unification and the simplifier, for debugging.\<close>
declare[[unify_trace_simp, unify_trace_types, simp_trace, simp_trace_depth_limit=1]]
section \<open>Functions\<close>
subsection \<open>Typing functions\<close>
text "Declaring \<open>Prod_intro\<close> with the \<open>intro\<close> attribute (in HoTT.thy) enables \<open>standard\<close> to prove the following."
lemma "\<^bold>\<lambda>x:A. x : A\<rightarrow>A" ..
proposition "A \<equiv> B \<Longrightarrow> \<^bold>\<lambda>x:A. x : B\<rightarrow>A"
proof -
assume assm: "A \<equiv> B"
have id: "\<^bold>\<lambda>x:A. x : A\<rightarrow>A" ..
from assm have "A\<rightarrow>A \<equiv> B\<rightarrow>A" by simp
with id show "\<^bold>\<lambda>x:A. x : B\<rightarrow>A" ..
qed
proposition "\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x : A\<rightarrow>B\<rightarrow>A"
proof
fix a
assume "a : A"
then show "\<^bold>\<lambda>y:B. a : B \<rightarrow> A" ..
qed
subsection \<open>Function application\<close>
proposition "a : A \<Longrightarrow> (\<^bold>\<lambda>x:A. x)`a \<equiv> a" by simp
text "Currying:"
lemma "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. y)`a`b \<equiv> b" by simp
lemma "a : A \<Longrightarrow> (\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). f x y)`a \<equiv> \<^bold>\<lambda>y:B(a). f a y" by simp
lemma "\<lbrakk>a : A; b : B(a); c : C(a)(b)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). \<^bold>\<lambda>z:C(x)(y). f x y z)`a`b`c \<equiv> f a b c" by simp
proposition wellformed_currying:
fixes
A::Term and
B::"Term \<Rightarrow> Term" and
C::"Term \<Rightarrow> Term \<Rightarrow> Term"
assumes
"A : U" and
"B: A \<rightarrow> U" and
"\<And>x::Term. C(x): B(x) \<rightarrow> U"
shows "\<Prod>x:A. \<Prod>y:B(x). C x y : U"
proof (rule Prod_formation)
fix x::Term
assume *: "x : A"
show "\<Prod>y:B(x). C x y : U"
proof (rule Prod_formation)
show "B(x) : U" using * by (rule assms)
qed (rule assms)
qed (rule assms)
proposition triply_curried:
fixes
A::Term and
B::"Term \<Rightarrow> Term" and
C::"[Term, Term] \<Rightarrow> Term" and
D::"[Term, Term, Term] \<Rightarrow> Term"
assumes
"A : U" and
"B: A \<rightarrow> U" and
"\<And>x y::Term. y : B(x) \<Longrightarrow> C(x)(y) : U" and
"\<And>x y z::Term. z : C(x)(y) \<Longrightarrow> D(x)(y)(z) : U"
shows "\<Prod>x:A. \<Prod>y:B(x). \<Prod>z:C(x)(y). D(x)(y)(z) : U"
proof (rule Prod_formation)
fix x::Term assume 1: "x : A"
show "\<Prod>y:B(x). \<Prod>z:C(x)(y). D(x)(y)(z) : U"
proof (rule Prod_formation)
show "B(x) : U" using 1 by (rule assms)
fix y::Term assume 2: "y : B(x)"
show "\<Prod>z:C(x)(y). D(x)(y)(z) : U"
proof (rule Prod_formation)
show "C x y : U" using 2 by (rule assms)
show "\<And>z::Term. z : C(x)(y) \<Longrightarrow> D(x)(y)(z) : U" by (rule assms)
qed
qed
qed (rule assms)
lemma curried_type:
fixes
a b A::Term and
B::"Term \<Rightarrow> Term" and
f C::"[Term, Term] \<Rightarrow> Term"
assumes "\<And>x y::Term. \<lbrakk>x : A; y : B(x)\<rbrakk> \<Longrightarrow> f x y : C x y"
shows "\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). f x y : \<Prod>x:A. \<Prod>y:B(x). C x y"
proof
fix x::Term
assume *: "x : A"
show "\<^bold>\<lambda>y:B(x). f x y : \<Prod>y:B(x). C x y"
proof
fix y::Term
assume **: "y : B(x)"
show "f x y : C x y" using * ** by (rule assms)
qed
qed
text "Note that the propositions and proofs above often say nothing about the well-formedness of the types, or the well-typedness of the lambdas involved; one has to be very explicit and prove such things separately!
This is the result of the choices made regarding the premises of the type rules."
text "The following shows that the dependent sum inductor has the type we expect it to have:"
lemma
assumes "C: \<Sum>x:A. B(x) \<rightarrow> U"
shows "indSum(C) : \<Prod>f:(\<Prod>x:A. \<Prod>y:B(x). C((x,y))). \<Prod>p:(\<Sum>x:A. B(x)). C(p)"
proof -
define F and P where
"F \<equiv> \<Prod>x:A. \<Prod>y:B(x). C((x,y))" and
"P \<equiv> \<Sum>x:A. B(x)"
have "\<^bold>\<lambda>f:F. \<^bold>\<lambda>p:P. indSum(C)`f`p : \<Prod>f:F. \<Prod>p:P. C(p)"
proof (rule curried_type)
fix f p::Term
assume "f : F" and "p : P"
with assms show "indSum(C)`f`p : C(p)" unfolding F_def P_def ..
qed
then show "indSum(C) : \<Prod>f:F. \<Prod>p:P. C(p)" by simp
qed
text "Polymorphic identity function."
consts Ui::Term
definition Id where "Id \<equiv> \<^bold>\<lambda>A:Ui. \<^bold>\<lambda>x:A. x"
(* Have to think about universes... *)
section \<open>Nats\<close>
text "Here's a dumb proof that 2 is a natural number."
proposition "succ(succ 0) : Nat"
proof -
have "0 : Nat" by (rule Nat_intro1)
from this have "(succ 0) : Nat" by (rule Nat_intro2)
thus "succ(succ 0) : Nat" by (rule Nat_intro2)
qed
text "We can of course iterate the above for as many applications of \<open>succ\<close> as we like.
The next thing to do is to implement induction to automate such proofs.
When we get more stuff working, I'd like to aim for formalizing the encode-decode method to be able to prove the only naturals are 0 and those obtained from it by \<open>succ\<close>."
end
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